The argument S = 1 - 1 + 1 - 1 + 1 - ...
then S = 1 - (1 - 1 + 1 - 1 + ...) = 1 - S,
S = 1 - S
S = 1/2 is invalid because S + S (from adding the terms) is the same as S. From this, it seems as though algebraic operations such as addition and multiplication by scalars do not seem to work the same way with infinite series as they do with numbers.

What struck me as odd was that even though infinite series, such as these, consist of algebraic operations on Real Numbers (integers in this case), algebraic operations do not seem to work on them in the same way...why is that so?

Even when the series is conditionally convergent, such as the harmonic series [itex]\sum_{n=1}^{\infty} \frac{1}{n}[/itex], there are still problems with associativity and commutativity of addition and distribution.

There is a theorem (which may be called Riemann's Rearrangement Theorem) that says for any conditionally convergent series there exists a permutation of the terms that will converge to any finite number desired or diverge to infinity or minus infinity.

The only series where the normal algebraic properties hold are those that are absolutely convergent.